Umbilical points on three dimensional strictly pseudoconvex CR manifolds I: manifolds with U(1)-action

A family of compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^2$ without umbilical points

“…However, they are not complete circular, and no circle action on D ǫ is everywhere transversal (to the CR structure). Thus, Theorem 1.1 does not contradict the existence results in [6].…”

The Log term in the Bergman and Szeg\H o kernels in strictly pseudoconvex domains in $\mathbb C^2$

“…Proof. The first identity in (62) was already observed in [ED15], Proposition 4.1 (but note that in that paper the complex conjugate of Q was considered). The proof is a direct computation of K ;zz , using the expressions By the first identity in (62), we have By expanding this, comparing with (44) and recalling b = 2Dφ, we conclude that the second identity in (62) holds.…”

Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in $\mathbb C^2$

“…For perturbations M ε of the sphere, as in (6.1), it is straightforward to verify that the M ε are circular for all sufficiently small ε > 0 if and only if in the decomposition (6.8) we have ρ ′ p,q = 0 for |p − q| = 0. It was shown in [6] that compact, circular real hypersurfaces in C 2 always have umbilical points. Here we shall consider perturbations M ε that are almost circular, which we define to be those for which, in the decomposition (6.8) of ρ ′ , we have ρ ′ p,q = 0 when |p − q| ≥ 4; we also say that such ρ ′ are almost circular.…”

“…We point out the analogy with the classical Caratheodory Conjecture (see, e.g., [10], [13]) regarding umbilical points on compact surfacces embedded in R 3 , and refer the interested reader to the paper [6] for a closer discussion of the analogy between the notion of (CR) umbilical points in CR geometry and that of umbilical points in the classicial geometry of surfaces in R 3 .…”

“…It was shown by X. Huang and S. Ji [11] that every real ellipsoid in C 2 must have umbilical points. More recently, it was proved by the first author and S. Duong [6] that every circular M 3 ⊂ C 2 has umbilical points; in fact, it was proved in [6] that every compact three dimensional CR manifold with a transverse free CR U(1) (circle) action must have umbilical points provided that the Riemann surface M/U(1) has genus g = 1.…”